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- Alternating_factorial abstract "In mathematics, an alternating factorial is the absolute value of the alternating sum of the first n factorials.This is the same as their sum, with the odd-indexed factorials multiplied by −1 if n is even, and the even-indexed factorials multiplied by −1 if n is odd, resulting in an alternation of signs of the summands (or alternation of addition and subtraction operators, if preferred). To put it algebraically,or with the recurrence relationin which af(1) = 1.The first few alternating factorials are1, 1, 5, 19, 101, 619, 4421, 35899, 326981, 3301819, 36614981, 442386619, 5784634181, 81393657019 (sequence A005165 in OEIS)For example, the third alternating factorial is 1! − 2! + 3!. The fourth alternating factorial is −1! + 2! - 3! + 4! = 19. Regardless of the parity of n, the last (nth) summand, n!, is given a positive sign, the (n - 1)th summand is given a negative sign, and the signs of the lower-indexed summands are alternated accordingly.This pattern of alternation ensures the resulting sums are all positive integers. Changing the rule so that either the odd- or even-indexed summands are given negative signs (regardless of the parity of n) changes the signs of the resulting sums but not their absolute values.Miodrag Zivković proved in 1999 that there are only a finite number of alternating factorials that are also prime numbers, since 3612703 divides af(3612702) and therefore divides af(n) for all n ≥ 3612702. As of 2006, the known primes and probable primes are af(n) for (sequence A001272 in OEIS)n = 3, 4, 5, 6, 7, 8, 10, 15, 19, 41, 59, 61, 105, 160, 661, 2653, 3069, 3943, 4053, 4998, 8275, 9158, 11164 Only the values up to n = 661 have been proved prime in 2006. af(661) is approximately 7.818097272875 × 101578.".
- Alternating_factorial wikiPageExternalLink wa.exe?A2=ind0411&L=nmbrthry&T=0&P=1106.
- Alternating_factorial wikiPageExternalLink lfact.pdf.
- Alternating_factorial wikiPageID "3484419".
- Alternating_factorial wikiPageRevisionID "605427330".
- Alternating_factorial hasPhotoCollection Alternating_factorial.
- Alternating_factorial title "Alternating Factorial".
- Alternating_factorial urlname "AlternatingFactorial".
- Alternating_factorial subject Category:Factorial_and_binomial_topics.
- Alternating_factorial subject Category:Integer_sequences.
- Alternating_factorial type Abstraction100002137.
- Alternating_factorial type Arrangement107938773.
- Alternating_factorial type Group100031264.
- Alternating_factorial type IntegerSequences.
- Alternating_factorial type Ordering108456993.
- Alternating_factorial type Sequence108459252.
- Alternating_factorial type Series108457976.
- Alternating_factorial comment "In mathematics, an alternating factorial is the absolute value of the alternating sum of the first n factorials.This is the same as their sum, with the odd-indexed factorials multiplied by −1 if n is even, and the even-indexed factorials multiplied by −1 if n is odd, resulting in an alternation of signs of the summands (or alternation of addition and subtraction operators, if preferred).".
- Alternating_factorial label "Alternating factorial".
- Alternating_factorial label "交互階乗".
- Alternating_factorial sameAs 交互階乗.
- Alternating_factorial sameAs m.09fzpw.
- Alternating_factorial sameAs Q3890817.
- Alternating_factorial sameAs Q3890817.
- Alternating_factorial sameAs Alternating_factorial.
- Alternating_factorial wasDerivedFrom Alternating_factorial?oldid=605427330.
- Alternating_factorial isPrimaryTopicOf Alternating_factorial.